Abstract

It is shown that every asymptotically regular or λ-firmly
nonexpansive mapping T:C→C has a fixed point
whenever C is a finite union of nonempty weakly compact convex
subsets of a Banach space X which is uniformly convex in every
direction. Furthermore, if {Ti}i∈I is any compatible family of strongly nonexpansive self-mappings on such a C and the graphs of Ti, i ∈I, have a nonempty intersection, then Ti, i∈I, have a common fixed point in C.

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